*In our previous blog post we had described a generic **Algorithmic architecture for a BMS** that attempts to strike a balance between software maintenance, performance and robustness. In this post we focus on another aspect of BMS algorithm design, namely the characterisation of cell parameters. Estimation of cell parameters is the fundamental building block of any algorithm used in the BMS. The Kalman Filter, the charging and discharge controllers, charging voltage controls etc.. all require knowledge of one or more cell parameters to operate effectively.*

If we were to survey the literature for cell models, we would find several models both at the electrolyte/electrode level as well as equivalent circuit /empirical models that approximate the dynamic behaviour of a Li-Ion cell. However while there are descriptions of how to extract the cell parameters from various experiments, no single reference provides a unified guide to extract the cell parameters. In this blog post, we will describe the sequence of steps that are required to extract the cell parameters as well as an example of how to perform this extraction.

### Equivalent Circuit Models

Like in any system, modeling of a Lithium ion cell can be done in broadly 3 different ways — **White box modeling, Grey box modeling and Black box modeling.** White box modeling for a system as complex as an electrochemical cell is extremely hard. It is testing intensive and requires information down to the levels of the quantity and grade of materials used in the cell. Black box modeling lies on the other end of the spectrum where it treats the model purely as a mathematical problem of curve fitting. While this is also testing intensive, the main drawback of this is that at the completion of the curve fitting exercise, you will end up with a black box without any way to infer the impact of change of any parameter not modeled explicitly as inputs or outputs on the system. This is where a Grey box model comes in very handy with a theoretical structure complemented by data to complete the model.

Within the paradigm of Grey box models for Lithium-Ion cells, there are a wide range of models like the **Shepherd Model, Unnewehr Universal Model, Nernst Model and the Equivalent Circuit Model.** While each of them have their pros and cons vis-à-vis complexity, accuracy, requirements of test data, physical intuition etc., one feature of the Equivalent Circuit Model that we particularly like is the linear ordinary differential equation structure that significantly aids in the process of controller synthesis for charge and discharge controls. Therefore, we recommend the use of the Equivalent Circuit Model for the cell. The equations that define this model are also shown below.

The advantage of a cell model like this is it is generically extensible till the model reaches acceptable performance. Figure 2 demonstrates this. The measured terminal voltage response of a cell for a discharge current pulse is shown by the blue trace, the dashed lines show approximations starting from 0 RC pairs to 6 RC pairs. As we increase the number of RC pairs the response becomes more and more aligned with the measured response. In fact there is virtually no difference between the measured response and the 6 RC pair model response. In fact, even the 2 RC pair model (red trace) could perhaps be an acceptable compromise between the complexity of the model and accuracy. For offline analysis the complexity of the model is not a barrier as compute time available is virtually limitless. However in the scenario where we have to use any of these models in an online iterative algorithm such as a Kalman Filter is where the complexity vs accuracy trade-off needs to be evaluated.

There are of-course drawbacks to the RC parameter model. For example, hysteresis in VOC-SOC table cannot be accurately captured. Coulombic efficiency is similarly hard to model without losing the linearity of the state equation. Despite these drawbacks, the RC model finely treads the line between categories: theoretically accurate but impossible to implement vs. so simple that it does not realistically predict anything. One important point to note here is that the requirements of linearity and ODE structure for the model are mainly for the purpose of observer design and controller synthesis i.e. the Kalman filter and discharge and charge controls. This does not limit one from using a more accurate and computationally complex model as the ‘Plant’ to perform closed loop simulations during the design of the aforementioned observer and controller or for other performance simulations.

### Experimental Cell Characterisation

Experimental data under controlled conditions is a crucial part of building the equivalent circuit model. Below we describe the experiments required to derive data points concerning three parameters described in the model. namely, the cell charge capacity (Q), the VOC-SOC curve and the HPPC test to derive the RC parameters.

**Cell Charge Capacity (Q), **generally measured in Ah, is the maximum amount of charge that can be extracted from cell under specific conditions. To measure the cell charge capacity, charge the cell to full. Follow this with a constant current discharge at Q*nom*/5 Amp till the voltage reaches the end of discharge voltage. Now integrate the value of current through the discharge cycle to get Q (refer Fig 3).

Information on conditions for ‘full charge’ and ‘end of discharge’ are typically provided in cell datasheets. Q*nom*/5 is the most commonly accepted industry standard for the measurement of charge capacity of Li-ion cells. An important point to note here is that while the nominal charge capacity Q*nom* of the Li-ion cell is provided in most datasheets, what we want to estimate here is the actual capacity Q from a statistically significant number of samples of cells.